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Culler and Vogtmann defined a simplicial space $O(g)$ called outer space tostudy the outer automorphism group of the free group $F_g$. Usingrepresentation theoretic methods, we give an embedding of $O(g)$ into theanalytification of $\mathcal{X}(F_g, SL_2(\mathbb{C})),$ the $SL_2(\mathbb{C})$character variety of $F_g,$ reproving a result of Morgan and Shalen. Then weshow that every point $v$ contained in a maximal cell of $O(g)$ defines a flatdegeneration of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ to a toric variety$X(P_{\Gamma})$. We relate $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ and $X(v)$topologically by showing that there is a surjective, continuous, proper map$\Xi_v: \mathcal{X}(F_g, SL_2(\mathbb{C})) \to X(v)$. We then show that thismap is a symplectomorphism on a dense, open subset of $\mathcal{X}(F_g,SL_2(\mathbb{C}))$ with respect to natural symplectic structures on$\mathcal{X}(F_g, SL_2(\mathbb{C}))$ and $X(v)$. In this way, we construct anintegrable Hamiltonian system in $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ for eachpoint in a maximal cell of $O(g)$, and we show that each $v$ defines atopological decomposition of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ derived fromthe decomposition of $X(v)$ by its torus orbits. Finally, we show that thevaluations coming from the closure of a maximal cell in $O(g)$ all arise asdivisorial valuations built from an associated projective compactification of$\mathcal{X}(F_g, SL_2(\mathbb{C})).$

Toric geometry of $SL_2(\mathbb{C})$ free group character varieties from outer space